Nov 10, 2012

Sam Loyd Puzzle Solvability

Problem: The15-puzzle(also calledGem Puzzle,Boss Puzzle,Game of Fifteen,Mystic Squareand many others) is asliding puzzlethat consists of a frame of numbered square tiles in random order with one tile missing. The object of the puzzle is to place the tiles in order (from figure at left to figure at right) by making sliding moves that use the empty space. Prove that the 15 puzzle in the configuration as shown on the left is not solvable.

Related Links:
I made this fb game yesterday night - 9 Puzzle over Movie Posters - Movie Sliding Puzzle
Sam Loyd published some great puzzles in his time. One book that gets recommended a lot is "Sam Loyd's Cyclopedia of 5000 Puzzles tricks and Conundrums" (Amazon , Flipkart)

2 comments:

here is solution.write the numbers horizontally so that it represents permutation of numbers 1-15(ignore the blank box).there are two different kind of slides possible.1.horizontal slide2.vertical slidehorizontal slide does not change the permutationeffect of vertical slide is change the position of a particular element by 3 units .for example in the figure 1 shown if 12 and the blank spot are swapped ,in the permutation position of 12 is 15,and position of 13,15,14 decreases by 1 and rest unchanged.now let us consider the parity of of number of inversions in a permutation.initial permutation has odd number of inversions.final one has even number of permutations.only vertical slide toggles the parity.since the blank occupies same row in both configuration,number of vertical slides are even. that is parity of initial and final permutations can not be different.so not possible

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I am an early stage technology investor at Nexus Venture Partners. Prior to this, I was a 3x product entrepreneur. Prior to this, I worked as a private equity analyst at Blackstone and as a quant analyst at Morgan Stanley. I graduated from Department of Computer Science and Engineering of IIT Bombay. I enjoy Economics, Dramatics, Mathematics, Computer Science and Business.

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